Bhabha scattering in the gauge-Higgs unification

We examine effects of $Z'$ bosons in gauge-Higgs unification (GHU) models at $\mathrm{e}^{+}\mathrm{e}^{-}\to \mathrm{e}^{+}\mathrm{e}^{-}$ Bhabha scatterings. We evaluate differential cross sections in Bhabha scatterings including $Z'$ bosons in two types of $\mathrm{SO}(5) \times \mathrm{U}(1) \times \mathrm{SU}(3)$ GHU models. We find that deviations of differential cross sections in the GHU models from those in the SM can be seen at $\sqrt{s} = 250\,\mathrm{GeV}$. With $80\%$-longitudinally polarized electron and $30\%$-longitudinally polarized positron beams, the left-right asymmetries in the GHU A- and B-models are resolved at more than $3\,\sigma$ at $L_{\rm int} =250\,\mathrm{fb}^{-1}$. We also show that Bhabha scattering with scattering angle less than 100 mrad can be safely used as luminosity measurements in $\mathrm{e}^{+}\mathrm{e}^{-}$ colliders since the effects of $Z'$ bosons are well suppressed for small scattering angle. We propose a new observable which can be measured at future TeV-scale $\mathrm{e}^{+} \mathrm{e}^{-}$ linear colliders.

In e + e − collider experiments, effects of GHU can be examined by exploring interference effects among photon, Z boson and Z bosons. In the previous papers we have studied effects of new physics (NP) on such observable quantities as cross section, forwardbackward asymmetry and left-right asymmetry in e + e − → ff (f = e) with polarized and unpolarized e + e − beams [21,25,30,31,[49][50][51]. In Ref. [21] we compared such observable quantities of GHU with those of the SM in LEP experiments at √ s = M Z , and LEP2 experiments for 130GeV ≤ √ s ≤ 207 GeV [52,53]. In Refs. [21,25] we also gave predictions of several signals of Z bosons in GHU in e + e − collider experiments designed for future with collision energies √ s ≥ 250 GeV with polarized electron and positron beams.
In the e + e − → ff (f = e) modes, the deviations of total cross sections become large for right-polarized electrons in the A-model, whereas in the B-model the deviations are large for left-polarized electrons.
Deviations from the SM can be seen in the Higgs couplings as well. HW W , HZZ and Yukawa couplings deviate from those in the SM in a universal manner [16,17,32]. They are suppressed by a common factor for W and Z bosons, and for SM fermions f . In the analysis of both Z and W bosons in hadron colliders [19,20,28], it is found that the AB phase is constrained as θ H 0.  [47]. Since the masses and Higgs couplings of the SM fields in the GHU models are very close to those in the SM, the electroweak phase transition (EWPT) occurs at T C ∼ 160GeV and appears very weak first order [22,27] in both A-and B-models, which is very similar to EWPT in the SM [48].
In this paper we study effects of Z bosons in GHU models on the e + e − → e + e − Bhabha scattering. Measurements of Bhabha scattering at linear colliders have contributed to the establishment of the SM [56][57][58]. Bhabha scattering is also useful to explore NP [59,60].
Unlike e + e − → ff (f = e) scattering, in Bhabha scattering not only s-channel but also t-channel contributions enter the process. Since the t-channel contribution of photon exchange dominates in forward scattering amplitudes, the cross section becomes very large for small scattering angles, which improves the statistics of experiments. It will be seen below that effects of Z bosons on cross sections can be measured with large significances.
Bhabha scattering at very small scattering angles is used for the determination of the luminosity of the e + e − beams. Since cross sections of all other scattering processes depend on the luminosity, one needs to know whether or not effects of Z bosons on the e + e − → e + e − cross section are sufficiently small. Forward-backward asymmetry A F B of the cross section in Bhabha scattering is no longer a good quantity for searching NP, since the backward scattering cross section is much smaller than the forward scattering cross section. We will propose a new quantity A X to measure with polarized e + e − beams, which can be used for seeing NP effects instead of A F B .
In Section 2 we briefly review the GHU A-and B-models and discuss the e + e − → e + e − scattering in both the SM and GHU models. In Section 3, we show the formulas of e + e − → e + e − scattering cross sections for longitudinally polarized e ± beams, and numerically evaluate the effects of Z bosons in GHU models on differential cross sections and left-right asymmetries. We also show that effects of Z bosons on the cross section are very small at the very small scattering angle.

Gauge-Higgs unification
In GHU A-and B-models the electroweak SU(2)×U(1) symmetry is embedded in SO(5)× U(1) X symmetry in the Randall-Sundrum warped space [46], whose metric is given by where η µν = diag(−1, +1, +1, +1) and k is the AdS-curvature. We refer two 4D hyperplanes at z = 1 and z = z L as the UV and IR branes, respectively. The SO (5)  vector multiplets in the SO(11) gauge-Higgs grand unification [37,38]. We also note that the bulk mass parameter for the bulk electron field, c e , is positive in the A-model [18] whereas c e in the B-model has to be negative [23]. In the B-model positive c e leads to an exotic light neutrino state and therefore negative c e must be chosen. Zero modes of fermion fields with positive bulk mass parameters are localized near the UV brane whereas zero modes of fermion fields with negative bulk mass parameters are localized near the IR brane.  in the right-handed components of SU(2) R doublet (ν e , e ) [23].
Interactions of the electron and gauge bosons are given by    Table 3: Left-handed and right-handed couplings of the electron, R and γ (1) ), in units of Ratios of e and g w are shown in the second column.
In Tables 2 and 3, parameters and couplings in the A-and B-models are tabulated.
Here, model parameters (θ H , m KK and z L ), masses, widths and couplings of Z -bosons are referred from Refs. [25,26]. The big difference in the magnitude of z L in the Aand B-models originates from the formulas of top-quark mass. 23]. In both models the W boson mass is given by m W m KK /(π √ kL) sin θ H so that the lower bound of z L becomes z L 8 × 10 3 in the A-model and z L 7 × 10 7 in the Bmodel. In Table 2 the bulk mass parameter for the electron field c e is given. As explained before, c e is positive in the A-model whereas c e is negative in the B-model. In Table 3, the left-and right-handed electron couplings to Z bosons, L is the 4D gauge coupling of the SO (5) where g A is the 5D SO(5) coupling. In terms of g A and the 5D U(1) X coupling g B , a mixing parameter is defined as [19,25] e/g w = sin θ 0 The value of sin θ 0 W is determined so as to reproduce the experimental value of the forwardbackward asymmetry in e + e − → µ + µ − scattering at the Z-pole. In the A-model electron's right-handed couplings to Z -bosons are larger than left-handed couplings. In the B-model electron's left-handed couplings to Z -bosons are larger than right-handed couplings.
3 Bhabha scattering in e + e − colliders We consider the e + e − → e + e − scattering in the center-of-mass frame. In this frame, the Mandelstam variables (s, t, u) are given by where E is the energy of initial electron and positron, and θ is the scattering angle of the electron. Since the e + e − → e + e − scattering process consists both s− and t−channel processes, the scattering amplitude is written in terms of the following six building blocks: where M V i and Γ V i are the mass and width of the vector boson V i . V i and r V i are left-and right-handed couplings of electrons to the vector boson respectively. In particular, we have Here e, I 3 e and θ 0 W are the electromagnetic coupling, weak isospin of the electron and the bare Weinberg angle defined in (5), respectively.
When initial state electrons and positrons are longitudinally polarized, the differential cross section is given by where P e − and P e + are the polarization of the electron and positron beam, respectively. P e − = +1 (P e + = +1) denotes purely right-handed electrons (positrons) [54,55]. σ e − X e + Y (X, Y = L, R) denotes the cross section for left-handed or right-handed electron and positron. When the electron mass is neglected, these cross sections are given by When s, t M 2 Z , the cross section is approximated by the one at the QED level, where we obtain S LL = S RR = S LR = e 2 /s and T LL = T RR = T LR = e 2 /t, and For unpolarized electron or positron beams, the above expression reduces to a familiar formula dσ unpolarized QED d cos θ = e 4 16πs We also note that in terms of building blocks (7) we can write down components of s-, t-channels, and interference terms as where each component is given by dσ interference d cos θ (P e − , P e + ) = 1 16πs u 2 (1 + P e − )(1 − P e + )Re(S RR T * RR ) When initial electrons and/or positrons are longitudinally polarized, one can measure left-right asymmetries. The left-right asymmetry of polarized cross sections is given by A LR (P − , P + ) ≡ σ(P e − = −P − , P e + = −P + ) − σ(P e − = +P − , P e + = +P + ) σ(P e − = −P − , P e + = −P + ) + σ(P e − = +P − , P e + = +P + ) where the cross section in a given bin [θ 1 , θ 2 ] is given by σ ≡ We can also define the left-right asymmetry of the differential cross section as where we have used dσ e − L e + L /d cos θ = dσ e − R e + R /d cos θ and defined In linear colliders we can measure the cross sections for (P e − , P e + ) = (P − , P + ), (P − , −P + ), (−P − , P + ) and (−P − , −P + ). Combining these quantities, one can make a new observable quantity which does not depend on the value of P ± . In e + e − → e + e − scatterings we have A LR (P − , +P + , cos θ) and A LR (P − , −P + , cos θ) as independent observables and one may define the following non-trivial quantity: (P − − P + )A LR (P − , −P + , cos θ) − (P − + P + )A LR (P − , +P + cos θ) (P − − P + )A LR (P − , −P + , cos θ) + (P − + P + )A LR (P − , +P + , cos θ) .
where the second equality holds only when P ± = 0 and |P + | = |P − |. As is evident in the first line of (17), A X (cos θ) is independent of the magnitudes of polarization P ± . This quantity may be used to explore NP beyond the SM as discussed below.
Since e + e − → e + e − scattering contains t-channel processes, forward scatterings dominate. Therefore unlike the e + e − → ff (f = e − ) scattering, the forward-backward asymmetry of e + e − → e + e − scattering is a less-meaningful quantity.
We note that all of the above formulas can be applied to + − → + − ( = µ, τ ) scatterings.

Numerical Study
In the followings we calculate e + e − → e + e − cross sections both in the SM and GHU models, and we evaluate effects of Z bosons in GHU models on observables given in the previous section. As benchmark points, we have chosen typical parameters of the A-and B-models in Tables 2 and 3. For experimental parameters we choose √ s = 250 GeV and L int = 250 fb −1 as typical value of linear e + e − colliders like ILC [61]. We also choose L int = 2 ab −1 , which will be achieved at circular e + e − colliders like FCC-ee [62] and CEPC [63]. For the new asymmetry A X (cos θ) in (17), we consider √ s = 3 TeV for future linear colliders like CLIC [64]. As for the longitudinal polarization, we set the parameter ranges −0.8 ≤ P e − ≤ +0.8 and −0.3 ≤ P e + ≤ 0.3, which can be achieved at ILC [61].
In Figure 1 e + e − → e + e − differential cross sections in the SM are plotted. In the forward-scattering region (cos θ > 0), the magnitudes of cross sections of t-channel and the interference parts are much larger than those of the s-channel part.  In Figures 2 and 3, the differences of differential cross sections of the GHU from the SM for unpolarized and polarized beams are plotted, respectively. In the figures, differences of s-channel, t-channel and interference contributions are also plotted. In the s-channel, the NP effects contribute destructively in the forward scattering. On the other hand, in the t-channel NP effects contribute constructively. Since the cross section is dominated by t-channel, the total of s-, t-and interference channels increases due to the NP effects.
In the A-model Z bosons have larger couplings to right-handed electrons than to lefthanded electrons. Therefore the cross section of the e − R e + L initial states becomes larger than that of e − L e + R . On the other hand, in the B-model Z bosons have larger couplings to left-handed electrons than to right-handed electrons, and the cross section of e − L e + R initial states becomes larger.
NP effects become smaller when θ becomes smaller. The statistical uncertainty, however, also becomes small since the cross section becomes very large. Therefore deviations of the cross section relative to statistical uncertainties may become large.
For unpolarized e + e − beams (Figure 3), the new physics effect in both models tends to enhance the cross section at forward scattering with almost the same magnitude. In the B-model the suppression of NP effects due to larger Z masses is compensated by larger couplings of Z bosons than the couplings in the A-model. The enhancement of the differential cross section due to the NP effects at cos θ ∼ 0.3 is around 1 %.
For polarized beams deviations can be much larger. In the A-model, electrons have large right-handed couplings to Z bosons and for right-handed polarized beam relative deviations of the cross-section become as much as 2 % [ Figure 3- In Figure 4, the statistical significances in the GHU models are plotted. An estimated significance of the deviation of the cross section in a bin is given by where N GHU and N SM are observed numbers of events in a bin. In Figure 4, significances are larger than 5 σ for cos θ 0.1. Significances are very large for forward scatterings, but are very small for backward scatterings. In Figure 4, relative 0.1 % errors are also shown.
Errors due to the NP effects become very small and are around 0.1 % for cos θ 0.9. A similar analysis has been given in Ref. [59].
For small scattering angles θ, the scattering amplitude is dominated by t-channel contributions which are constructed with the blocks T LL , T RR and T LR . When |t| sθ 2 /4 M 2 Z , M 2 Z , we can approximate the SM and NP contributions to the block T LL in the scattering amplitude as When sθ 2 /4 M 2 Z , T LL is dominated by the QED part T QED LL −4e 2 /(θ 2 s) and the NP effects are estimated as and similar analysis is applied to T LR and T RR . Consequently, this correction arises not only in amplitudes but also in differential cross sections. For Unpolarized, s =250 GeV When the initial electron and positron beams are longitudinally polarized, the leftright asymmetry A LR (14) can be measured. In Figure 6, the left-right asymmetries of the SM and GHU models are plotted. The measured asymmetries become larger when |P e − − P e + | are larger. As seen in Figure 2, in the A-model cross section of e − R e + L initial states becomes large whereas in the B-model cross section of e − L e + R initial states is enhanced due to the large left-handed Z couplings. Therefore A LR of B-models are larger than the SM, whereas A LR of A-models are smaller. Since the A LR is proportional to |P e − − P e + |, the asymmetries in Figure 6-(a) are almost twice as large as in (b). In Figure 6, an asymmetry A LR in a bin and the statistic error ∆A LR are also shown. Here with N L and N R being observed number of events for the left-handed (P e − < 0) and righthanded (P e − > 0) electron beams, respectively. For small scattering angle cos θ 0.8, both A GHU LR and A SM LR become close to each other. To see how NP effects against statistical uncertainty grow for small θ, we plotted in For the forward scattering with cos θ 0.2, the deviations are bigger than several times of standard deviations.  In Figure 8, the asymmetry A X defined in Eq. (17) is plotted for √ s = 250 GeV and √ s = 3 TeV. At √ s = 250 GeV, the NP effect on A X is very small. For √ s = 3 TeV, the asymmetry A X of the SM and GHU models is clearly different and may be discriminated experimentally. In the present analysis of NP effects, only first KK excited states of neutral bosons are taken into account. At √ s ∼ 3 TeV, effects of second KK modes on A X are estimated as a few percent. These effects are much smaller than the effects of the first KK modes.

Summary
In this paper we examined the effects of Z bosons in the gauge-Higgs unification (GHU) models in the e + e − → e + e − (Bhabha) scatterings. We first formulated differential cross sections in Bhabha scattering including Z bosons. We then numerically evaluated the deviations of differential cross sections in the two SO(5) × U(1) × SU(3) GHU models (the A-and B-models) at √ s = 250 GeV. We found that at L int = 2 ab −1 with unpolarized e + e − beams, the deviation due to Z bosons in the GHU models from the SM can be clearly seen. We also found that for 80 %-longitudinally polarized electron and 30 %polarized positron beams, deviations of the differential cross sections from the SM become as large as a few percent for cos θ ∼ 0.2, and that the A-model and the B-model are well distinguished at more than 3 σ significance at L int = 250 fb −1 . We also checked the effects of Z bosons are negligible for the scattering angle smaller than 100 mrad at √ s = 250 GeV. Therefore Bhabha scatterings at very small θ can be safely used as the measurements of the luminosity in the e + e − collisions. Finally we introduced the new observable A X . We numerically evaluated it at √ s = 250 GeV and 3 TeV. Effects of the GHU models on A X can be seen at future TeV-scale e + e − colliders.
In this paper the effects of Z bosons are calculated at the Born level. Higher-order QED effects should be taken into account for more precise evaluation [65,66].